Full solution

Q. Solve for

k

\newline

\log_{10}19k = 19\log_{10}k

Given Equation Simplification: We are given the equation $\log_{10}19k = 19\log_{10}k$ . We need to find the value of $k$ that satisfies this equation. $\newline$ First, let's use the power rule of logarithms, which states that $\log_b(m^n) = n\log_b(m)$ , to simplify the right side of the equation.
Logarithmic Equivalence: Applying the power rule to the right side of the equation, we get: $\newline$ $\log_{10}19k = \log_{10}(k^{19})$ . $\newline$ Now, since the bases of the logarithms on both sides of the equation are the same, we can equate the arguments of the logarithms.
Solving for $k$ : Setting the arguments of the logarithms equal to each other, we have: $\newline$ $19k = k^{19}$ . $\newline$ Now, we need to solve for $k$ .
Checking Solutions: To solve the equation $19k = k^{19}$ , we can see that $k = 0$ is not a solution since $19\times 0$ is not equal to $0^{19}$ . We can also see that $k = 1$ is a solution since $19\times 1$ is equal to $1^{19}$ . Let's check if there are any other solutions.
Final Solution: Dividing both sides of the equation by $k$ (assuming $k \neq 0$ ), we get: $\newline$ $19 = k^{18}$ . $\newline$ Now, we need to find the value of $k$ that satisfies this equation.
Final Solution: Dividing both sides of the equation by $k$ (assuming $k \neq 0$ ), we get: $\newline$ $19 = k^{18}$ . $\newline$ Now, we need to find the value of $k$ that satisfies this equation.The equation $19 = k^{18}$ can only be true if $k = 1$ , because $1$ raised to any power is still $1$ . Any other value of $k$ raised to the $18$ th power will not equal $k \neq 0$ $0$ .
Final Solution: Dividing both sides of the equation by $k$ (assuming $k \neq 0$ ), we get: $19 = k^{18}$ . Now, we need to find the value of $k$ that satisfies this equation.The equation $19 = k^{18}$ can only be true if $k = 1$ , because $1$ raised to any power is still $1$ . Any other value of $k$ raised to the $18$ th power will not equal $k \neq 0$ $0$ .Therefore, the only solution to the equation $k \neq 0$ $1$ is $k = 1$ .